Goal of the seminar will be to discuss general facts about generating functions and some of the classical (and more recent) applications.


18.07.2024   Luca Asselle "The contact non-squeezing theorem."    
Abstract: We prove a contact version of the celebrated non-squeezing theorem of Gromov. References: [4],[8].


11.07.2024   Pierre-Alexandre Arlove  "Uniqueness of gfqi."    
Abstract: We prove uniqueness (up to equivalence) of gfqi for Lagrangian submanifolds of T∗M resp. Legendrian submanifolds of J1(M, R). References: [6],[7].




04.07.2024   Jacobus de Pooter  "Existence of gfqi, Part 3."    
Abstract: In this talk we prove the existence of gfqi for Legendrian submanifolds in spherizations of cotangent bundles. Reference: [5]. 




27.06.2024   Lucas Dahinden  "Existence of gfqi, Part 2."    
Abstract: In this talk we prove the existence of gfqi for Legendrian submanifolds of J1(M,R). Reference: [3].




13.06.2024   Lars Kelling  "Existence of gfqi, Part 1."    
Abstract: In this talk we prove the existence of gfqi for Lagrangian submanifolds of T∗M. Reference: [1].





16.05.2024   Alberto Abbondandolo  "Geodesics in the Hofer norm and gene- rating functions."    
Abstract: Reference: [2].





02.05.2024   Michael Vogel  "Generating functions quadratic at infinity (gfqi), Part 3."    
Abstract: This is the third of three talks in which we discuss general facts about gfqi: Spectral invariants; applications to the geometry of the Hamiltonian group; Viterbo capacity and Gromov’s non-squeezing theorem. Reference: [9], Sections 11,12,14. 


                             




25.04.2024   Jonas Fritsch  "Generating functions quadratic at infinity (gfqi), Part 2."    
Abstract: This is the second of three talks in which we discuss general facts about gfqi: Gfqi for Hamiltonian symplectomorphisms of R2n; composition formulas; equivalence of gfqi; the symplectic action. Reference: [9], Sections 7-10.
                             




18.04.2024   Paul Anton Wilke  "Generating functions quadratic at infinity (gfqi), Part 1."    
Abstract: This is the first of three talks in which we discuss general facts about gfqi: Symplectic and Hamiltonian diffeomorphisms; Lagrangian submanifolds; coisotropic submanifolds, characteristic foliations and symplectic reduction; generating functions for Lagrangian submanifolds. Reference: [9], Sections 2-5.
                             




11.04.2024   Stefan Nemirovski  "Introduction to generating functions."    
Abstract: This is an overview talk on the different flavours of generating functions and where to find them.
                             

The main references for the seminar are:
[1] M. Brunella - On a theorem of Sikorav, L’Ens. Math ́ematique 37 (1991), 83–87.
[2] M. Bialy, L. Polterovich - Geodesics of Hofer’s metric on the group of Hamiltonian diffeomorphisms, Duke Math. J. 76 (1994), 273–292.
[3] E. Ferrand - On a theorem of Chekanov, Symplectic singularities and geometry of gauge fields, Banach center Publications 39 (1997).
[4] M. Fraser, S. Sandon, B. Zhang - Contact non-squeezing at large scale via generating functions (2023), https://arxiv.org/pdf/2310.11993.pdf
[5] P. Pushkar - Chekanov-type theorem for spherized cotangent bundles (2021), arXiv:1602.08743v1
[6] D. Th ́eret - Utilisation des fonctions g ́en ́eratrices en g ́eom ́etrie symplectique globale, Ph.D. Thesis, Universit ́e Denis Diderot (1995).
[7] D. Th ́eret - A complete proof of Viterbo’s uniqueness theorem on generating functi- ons, Topology Appl. 96 (1999), 249–266.
[8] S. Sandon - Contact homology, capacity and non-squeezing in R2n ×S1 via generating functions, Ann. Inst. Fourier (Grenoble) 61 (2011), 145–185.
[9] S. Sandon - Generating functions in symplectic topology (2014), http://members.unine.ch/felix.schlenk/Santiago/cours.margherita.pdf

Other references are:
• F. Laudenbach, and J.-C. Sikorav, Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibr ́e cotangent, Invent. Math. 82 (1985), 349–357.
• J.C. Sikorav - Sur les immersions lagrangiennes dans un fibr ́e cotangent admettant une phase g ́en ́eratrice globale, C.R. Acad. Sci. Paris, S ́er. I Math. 302 (1986), 119–122.
• J.C. Sikorav, Probl ́emes d’intersections et de points fixes en g ́eom ́etrie hamiltonienne, Comment. Math. Helv. 62 (1987), 62–73.
• C. Viterbo - Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), 685–710.